Research Updates in Mathematics and Computer Science Vol. 9

This book chapter delves into the integrability of a nonlinear system characterized by the emergence of endemic Malaria, utilizing Prelle-Singer and Lie symmetry analysis techniques. The model involves three nonlinear differential equations representing interactions among susceptible humans, infected humans, and infected mosquitoes. Through examining the biological plausibility of the proposed model and scrutinizing the integrability of the nonlinear system, this study sheds light on its dynamics. Furthermore, it showcases the integrability of the model through the presentation of an explicit solution. Additionally, exact invariant solutions of the model are derived by employing the obtained infinitesimal generators and corresponding similarity reduction equations, enriching our comprehension of the system’s behaviour and potential strategies for combating Malaria.

In this paper, an M/M/1/N a single server finite capacity Markovian queuing model, encouraged arrivals with controllable arrival rate is considered. The term encouraged arrivals emerged from situation that a system experiences after release of offers and discounts by firms. Encouraged arrival is a new addition to existing customer behaviour in queuing theory. The steady state solutions of system size are derived explicitly. The analytical results are numerically illustrated and relevant conclusions are presented.

Minimizing the computational cost of polynomial evaluation is a main problem in Computational Science. We present a modification of Horner’s method to exploit its advantages in the evaluation of sparse polynomials. We also propose a polynomial partitioning that allows to perform Horner’s method in parallel. In addition, we provide an analysis of the numerical error between the proposed parallel method and the classic algorithm.

The study proposes an effective sentiment analysis recommender system framework using machine learning models. Recommender systems are used to build recommendations by processing information from actively gathered varied kinds of data. The data that is used for processing information depends upon the type of recommender system. In recent years, with the rapid growth of Internet technology, online shopping has become a rapid way for users to purchase and consume desired products. Tweet sentiment analysis is a product of the vast amount of user-generated content on social media platforms like Twitter. Sentiment analysis serves as the foundation for recommendation and decision support systems, and it is becoming a crucial tool on online platforms to extract user emotional state data and increase user happiness.  

For any \(\mathrm{n}\) number of coupled nonlinear partial differential equations for spherically symmetric field equations of the typer \({ }^2\left(\frac{\partial^2 \phi_j}{\partial r^2}-\frac{\partial^2 \phi_j}{\partial t^2}\right)=F \_i(\phi \mathrm{j})\), where \(\mathrm{j}=1,2, \ldots \mathrm{n}\), are the number of dependent variables and \(\mathrm{F} \mathrm{i}\left(\phi \_\mathrm{j}\right)\) are any functions of dependent variables \(\phi_j, \mathrm{j}=1,2, . . \mathrm{n}\). and free of independent variables \(\mathrm{r}\) and \(\mathrm{t}\) then a similarity variable is found as \(\mathrm{s}(\mathrm{r}, \mathrm{t})=\mathrm{r} /\left[\left(r^2-t^2\right)-\kappa t / \tau+\kappa^2 /\left(4 \tau^2\right)\right]\), where \(\kappa\) and \(\tau \neq 0\) are arbitrary integration constants. Using \(s(r, t)\) above coupled partial differential equations can be transformed into coupled ordinary differential equations. This result may reduce lengthy calculations for finding similarity transformations of coupled partial differential equations. Using this similarity variable two exact Dyon solutions of spherically symmetric Yang-Mills-Higg's field equations are found with 'circular functions.' For which known solutions are with hyperbolic functions.

The study introduces an innovative approach to manage inventory for deteriorating items by taking into account the permissible delays and default risk. This perspective offers fresh insights into a complex problem faced by many businesses. A credit period is often extended by suppliers to their clients in order to foster long-term relationships and ensure their survival in the business environment. This draws in new clients, increasing market demand. Conversely, the provider is exposed to default risk when a credit period is present. In this paper, an inventory model is developed that deals with credit period dependent quadratic demand and default risk associated with sales revenue. The Deterioration rate under the natural environment is also incorporated with the inventory model. This article discusses the seller's best choice for determining the customer's allowable credit period duration and the purchase amount. Concerns about environmental degradation are also taken into account while making purchases in order to maximize profits. A solution procedure is given for finding the optimal solution of total profit. Numerical example is given to show the effectiveness of the model. Finally, sensitivity analysis is carried out to explore the managerial implications. This study will help significantly the seller in setting optimal credit periods. In future, this research can be extended to study inventory models with stochastic demand. This paper can be extended if shortages are allowed. Different preservation technologies can also be incorporated to reduce deterioration and enhance environmental protection.

The basic problem of differential calculus is the problem of tangent lines and calculating the slope of the tangent line to the graph at a given point P and the less seemingly important problem of defining the vertical asymptote line and its derivative. The Logical Derivative makes it feasible to compute tangent line equations of vertical asymptote lines with the corresponding slope and direction of the asymptotes. L'hopitals indeterminate forms were used to evaluate Newton’s difference quotient and compute the logical derivative”. These are new derivatives developed using a method of direct proportions. By reversing the decrement and factoring it along with further analysis, derivatives derived are of the same dimension as their functions.

A shallow cloud of warm microphysics is used to analyze the subgrid variability of liquid water content. These turbulent flows are diagnosed by means of second-order moment transport equations for liquid water quantities. These equations show that liquid water variabilities are controlled by the production of the gradient of mean liquid water quantities and microphysical processes. The contributions (source or sink) of these different productions reflect the effects of the gradient of average quantities of liquid water content and microphysical processes on the evolution of liquid water variability. It emerges that cloud water variance is mainly produced by the cloud water gradient term and constantly destroyed by the effects of auto conversion and accretion processes. Inversely, the processes of rain droplet formation and growth contribute as the main source for the variance of precipitating water and the cloud water-precipitating water covariance. Microphysical depletion processes, notably cloud droplet evaporation and precipitable droplet sedimentation, act as sinks for liquid water variances and covariance. Finally, for rainwater variability, the gradient term may be less important, but it provides real support for the source or sink terms of microphysics. This work particularly highlights the subgrid variabilities associated with precipitating droplets. In particular, this work focuses on the subgrid variabilities associated with liquid water content. Incorporating a good parameterization of liquid water variances and cloud water-rainwater correlation in statistical schemes could improve rainwater formation, growth and loss processes in large-scale models.

Understanding the dynamics of model behavior in response to changes in input parameters is pivotal, especially within complex models featuring an array of input factors. Sensitivity Analysis (SA) serves as a fundamental methodology for elucidating and quantifying the uctuations in model behavior corresponding to variations in model input factors. In situations where models incorporate a wide range of input factors, identifying the most in uential variables is of utmost importance. Although the employment of graphical tools to encapsulate SA findings is gaining traction, it remains a relatively new and evolving approach. The advancement of graphical representations significantly enhances the understanding of SA outcomes. Within this work, an exploration into the efficacy of two modern graphical techniques, specifically star plots and dot charts, as tools for SA is undertaken. These visual aids provide a clear representation of key input factors, thereby accelerating the comprehension process. To showcase their utility in SA,
these techniques are applied to two distinct compartmental models elucidating the dynamics of the global carbon cycle.

The present investigation of a two-dimensional problem in “modified couple stress’s thermoelastic medium for a half-space is “established and state-space approach” followed by normal mode analysis has been applied to solve the problem. In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations.  In this chapter, a homogeneous isotropic modified couple stress thermoelastic half space for the Lord-Shulman [L-S] model and Green-Lindsay [G-L] model has been considered. The exact expressions for normal stress, tangential stress and couple stress are obtained. These quantities are calculated numerically and depicted graphically for a special model. A particular case of interest has also been deduced from the present investigation.